Coastline Ruler: how long is every country's coast?

How long is the coast of Britain? It depends on your ruler.

Walk a pair of dividers along a coast in equal steps and add them up. Use a shorter ruler and you catch more bays and headlands, so the coast gets longer. Lewis Fry Richardson spotted this in the 1950s, and in 1967 Benoit Mandelbrot turned it into the fractal dimension D: 1 for a smooth line, higher for a more crinkled one. We ran the divider walk on every country's coastline, with rulers from 1 to 500 km. Drag the slider and watch.

Heads-up: this ranking depends on how islands are counted. A long ruler steps straight over any island shorter than itself, so small islands drop out of the measurement. Our first version dealt with that clumsily, so we measured again more fairly: now each island only counts at ruler sizes it is big enough for. Guinea-Bissau is still on top, but some countries moved a lot. What changed and why

What we found

7.5×Norway's whole coast measures 2,816 km with a 200 km ruler and 21,158 km with a 5 km one. Mainland Norway with its nearby islands (no Svalbard) grows 7.0×, from 2,313 km to 16,303 km.

1.507The most crinkled coast of the 89 countries we could rank is Guinea-Bissau, a mainland of drowned river valleys. (Its Bijagós islands are too small to measure fairly, so they are left out.) Then the French Southern Lands (1.430) and Norway (1.414).

1.020Libya and Namibia tie for straightest, then Morocco (1.034). Namibia's coast only grows from 1,367 to 1,463 km.

1.269Great Britain's D in our walk. Richardson's data gave about 1.25. We get 1.153 for mainland Australia (about 1.13 published) and 1.052 for South Africa (about 1.02).

The top of the table. After Guinea-Bissau (1.507) come the French Southern Lands (1.430), Norway (1.414), the Falkland Islands (1.413), Iceland (1.376), Greenland (1.367) and Bangladesh (1.367), whose Ganges delta and Sundarbans shred the coast into creeks. Fjords, glaciers and deltas all do the same job.

Mangroves match Britain. Nigeria (1.297) and Gabon (1.285) sit level with the United Kingdom (1.286) and ahead of Canada (1.272). Only Chile (1.300) edges past them. The Niger Delta's maze of channels is as rough at these scales as the Scottish lochs.

Correction: our first ranking counted islands badly. An island shorter than the ruler simply disappears from the walk. Add up a whole country and its total shrinks extra fast as the ruler grows, so countries with lots of islands look more crinkled than they are. (French Polynesia came out at an impossible 2.27 at first.) Our first fix kept only coast pieces at least 10 times longer than the biggest ruler, 100 km, and threw the rest away. That avoided the impossible numbers, but it ranked most countries on just their longest coasts: a median of 74% of each coast, 28% for Fiji, 32% for Croatia.

What we do now. Every coast piece (a mainland stretch or an island) is measured on its own, and only with rulers no longer than a twentieth of its length. A 1,000 km island counts up to a 50 km ruler, a 300 km island up to 15 km. Pieces shorter than 300 km can't be measured with four or more of our ruler sizes, so they are left out. A country's D is the average of its pieces' D, weighted by length. We tried a tenth first: 7 of 572 pieces then gave impossible values (up to 2.01), because on thin islands and peninsulas the ruler hops straight across. A twentieth gave none. This covers a median of 82% of each coast.

What moved. Guinea-Bissau stays on top, and the same three countries are still the straightest. The United Kingdom climbs from 21st to 13th (1.264 to 1.286), Estonia from 46th to 21st and Spain from 73rd to 52nd. Fiji falls from 9th to 49th (1.298 to 1.146): its old score rested on its biggest island alone (742 km of coast), measured with rulers up to 70 km. Up to a twentieth of its length that island scores 1.185, and the next island, at 1.093, pulls the average down further. Greece drops to 1.233 and Croatia moves to 1.135, but most of their coast is islands under 300 km, so even now they are only partly measured (54% and 32%). French Polynesia and Kiribati have no piece long enough, so they are no longer ranked. Some of the reshuffle isn't about islands at all: a piece shorter than 2,000 km is now fitted on a shorter range of rulers, and D depends on the range (see caveats). Guinea-Bissau went from 1.482 to 1.507 for exactly that reason. The table can still show the first version, and a "main coast only" version that measures just each country's longest connected coast piece.

Size doesn't decide it. Measured with a 5 km ruler, Canada's coast is 113,245 km and the USA's is 54,393 km. Their D values are almost the same, 1.272 and 1.266. Australia's coast only roughly doubles between the 200 km and 5 km rulers (14,032 to 29,686 km).

Every country, ranked by D

The 89 countries with at least 1,000 km of coast (measured along the full-detail Natural Earth line, before any ruler) and at least one coast piece long enough for a fit. "Move" is the change in rank from our first version. Click or press Enter on a row to load it.

#	Country	D	Move	5 km ruler	200 km ruler	In fit

Caveats

The map has a resolution limit. Natural Earth's 1:10m coastline has a point about every 2.2 km on average. With rulers shorter than that, the walk just follows the straight segments between points, so the curve flattens out. We fit D only on rulers from 5 to 100 km (shorter for short coast pieces) and shade the part below 5 km on the plot. A survey-grade coastline would keep growing below 5 km.

D depends on the range of scales. Richardson's rulers ran from about 10 to 1,000 km, and he measured the west coast of Britain, not the whole island. Fit Great Britain on 2–200 km and we get 1.245, on 10–200 km 1.290. Treat the third decimal as noise and the first as real.

Countries aren't natural objects. A country's coast is cut where its land borders meet the sea, and every piece is walked separately. Overseas territories count for the country Natural Earth assigns them to (France includes its overseas regions, for example). The islands list measures 39 single landmasses with no borders to worry about.

Lakes and rivers. We use Natural Earth's ocean coastline only. Estuaries are cut wherever Natural Earth's cartographers closed them, which matters a lot for deltas.

The map in your browser uses a lightly simplified copy of the coast (within 200 m), so its live length can differ by a fraction of a percent from the full-resolution numbers in the table and plot.

Sources and method

L. F. Richardson, "The problem of contiguity: an appendix to Statistics of Deadly Quarrels", General Systems Yearbook 6 (1961), 139–187. This is where the ruler-length rule comes from.

B. Mandelbrot, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", Science 156 (1967), 636–638. Uses Richardson's data and gives D ≈ 1.25 for the west coast of Britain, ≈ 1.13 for Australia and ≈ 1.02 for South Africa.

Coastlines: Natural Earth 1:10m admin 0 countries, coastline and land (v5, public domain). A country's coast is every edge of its polygon whose two ends are both points on the Natural Earth coastline.

Method: a geodesic divider walk on a sphere with radius 6,371 km. From each point, the next step lands at the first place further along the coast that is exactly one ruler length away in a straight line. The leftover bit at the end of each piece is added on. 17 rulers from 1 to 500 km. D = 1 − slope of log(length) against log(ruler), fitted by least squares. The code is a short Python script with no GIS libraries, and this page runs the same walk live in your browser.